Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 120277 by bemath last updated on 30/Oct/20

$$\underset{x\to \mathrm{\infty }}{\lim }{x}^3\{\sqrt{x^2+\sqrt{x^4+1}}-x\sqrt{2}\}?$$

Commented by benjo_mathlover last updated on 30/Oct/20

$$\underset{x\to \mathrm{\infty }}{\lim }{x}^4\{\sqrt{1+\sqrt{1+\frac{1}{x^4}}}-\sqrt{2}\}$$$$setting\frac{1}{x}=z\wedge z\to 0$$$$\underset{z\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+z^4}}-\sqrt{2}}{z^4}\times \frac{\sqrt{1+\sqrt{1+z^4}}+\sqrt{2}}{\sqrt{1+\sqrt{1+z^4}}+\sqrt{2}}$$$$\underset{z\to 0}{\lim }\frac{\sqrt{1+z^4}-1}{z^4}\times \frac{1}{2\sqrt{2}}=\frac{\sqrt{2}}{4}\times \underset{z\to 0}{\lim }\frac{z^4}{z^4(\sqrt{1+z^4}+1)}$$$$=\frac{\sqrt{2}}{4}\times \frac{1}{2}=\frac{\sqrt{2}}{8}$$

Answered by Dwaipayan Shikari last updated on 30/Oct/20

$$x^3(\sqrt{x^2+\sqrt{x^4+1}}-x\sqrt{2})$$$$=x^3(\sqrt{x^2+x^2\sqrt{1+\frac{1}{x^4}}}-x\sqrt{2})$$$$=x^3(x\sqrt{1+\sqrt{1+\frac{1}{x^4}}}-x\sqrt{2})$$$$=x^4(\sqrt{1+1+\frac{1}{2x^4}}-\sqrt{2})$$$$=\sqrt{2}{x}^4(\sqrt{1+\frac{1}{4x^4}}-1)$$$$=\sqrt{2}{x}^4(1+\frac{1}{8x^4}-1)$$$$=\frac{1}{4\sqrt{2}}$$

Answered by bemath last updated on 30/Oct/20

$$\underset{x\to \mathrm{\infty }}{\lim }{x}^3\left\{\frac{x^2+\sqrt{x^4+1}-2x^2}{\sqrt{x^2+\sqrt{x^4+1}}+x\sqrt{2}}\right\}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{x^3\{\sqrt{x^4+1}-x^2\}}{\sqrt{x^2+\sqrt{x^4+1}}+x\sqrt{2}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{x^3\{x^4+1-x^4\}}{x(\sqrt{1+\sqrt{1+\frac{1}{x^4}}}+\sqrt{2})(\sqrt{x^4+1}+x^2)}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{x^3}{x^3(\sqrt{1+\sqrt{1+\frac{1}{x^4}}}+\sqrt{2})(\sqrt{1+\frac{1}{x^4}}+1)}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{(\sqrt{1+\sqrt{1+\frac{1}{x^4}}}+\sqrt{2})(\sqrt{1+\frac{1}{x^4}}+1)}=$$$$\frac{1}{(\sqrt{2}+\sqrt{2})\times 2}=\frac{1}{4\sqrt{2}}$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com